- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Four-dimensional stationary algebraically special solutions, Weyl invariants, and soft hairs beyond large gauge transformations

H. Lü1,2,3,* and Pujian Mao1,†

  • *Contact author: mrhonglu@gmail.com
  • †Contact author: pjmao@tju.edu.cn
PDF

Phys. Rev. D 113, 024037 – Published 20 January, 2026

DOI: https://doi.org/10.1103/hx3b-8t9g

Abstract

We revisit the Ricci-flat metrics in four dimensions that are stationary and algebraically special, together with the locally asymptotically flat conditions in the generalized Bondi-Sachs framework. We show that the Einstein equation is reduced to the Laplacian equation on the celestial sphere. The solutions consist of two pairs of arbitrary holomorphic and antiholomorphic functions analogous to the Virasoro modes. We prove that the higher modes of one pair of the (anti)holomorphic function contain an infinite tower of soft hairs from the perspectives of the asymptotic supertranslation charges. We verify that different modes of the soft hairs are distinct solutions that cannot be connected by diffeomorphism, using the Weyl invariants associated to the solutions. Extending the construction to Einstein-Maxwell theory introduces a third pair of (anti)holomorphic functions, arising from the Maxwell tensor, which generates soft electric hair. We further present an exact soft-hairy solution of Einstein-Maxwell theory with a cosmological constant, offering a potential understanding of soft hair from the AdS/CFT correspondence.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (87)

  1. R. Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14, 57 (1965).
  2. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2023).
  3. K. Akiyama et al. (Event Horizon Telescope), First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole, Astrophys. J. Lett. 875, L1 (2019).
  4. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  5. S. W. Hawking, Black hole explosions, Nature (London) 248, 30 (1974).
  6. S. W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14, 2460 (1976).
  7. S. D. Mathur, The information paradox: A pedagogical introduction, Classical Quantum Gravity 26, 224001 (2009).
  8. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93, 035002 (2021).
  9. S. Raju, Lessons from the information paradox, Phys. Rep. 943, 1 (2022).
  10. D. N. Page, Information in black hole radiation, Phys. Rev. Lett. 71, 3743 (1993).
  11. A. Perez, Black holes in loop quantum gravity, Rep. Prog. Phys. 80, 126901 (2017).
  12. O. Lunin and S. D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B623, 342 (2002).
  13. S. D. Mathur, The fuzzball proposal for black holes: An elementary review, Fortschr. Phys. 53, 793 (2005).
  14. K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rep. 467, 117 (2008).
  15. A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, Black holes: Complementarity or firewalls?, J. High Energy Phys. 02 (2013) 062.
  16. G. Penington, Entanglement wedge reconstruction and the information paradox, J. High Energy Phys. 09 (2020) 002.
  17. A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, J. High Energy Phys. 12 (2019) 063.
  18. G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole interior, J. High Energy Phys. 03 (2022) 205.
  19. A. Almheiri, R. Mahajan, and J. Maldacena, Islands outside the horizon, arXiv:1910.11077.
  20. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, J. High Energy Phys. 05 (2020) 013.
  21. S. W. Hawking, M. J. Perry, and A. Strominger, Soft hair on black holes, Phys. Rev. Lett. 116, 231301 (2016).
  22. H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. R. Soc. A 269, 21 (1962).
  23. R. K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. R. Soc. A 270, 103 (1962).
  24. W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164, 1776 (1967).
  25. B. Carter, Axisymmetric black hole has only two degrees of freedom, Phys. Rev. Lett. 26, 331 (1971).
  26. D. C. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Lett. 34, 905 (1975).
  27. S. W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25, 152 (1972).
  28. P. T. Chrusciel, J. Lopes Costa, and M. Heusler, Stationary black holes: Uniqueness and beyond, Living Rev. Relativity 15, 7 (2012).
  29. G. Compère and J. Long, Vacua of the gravitational field, J. High Energy Phys. 07 (2016) 137.
  30. G. Compère and J. Long, Classical static final state of collapse with supertranslation memory, Classical Quantum Gravity 33, 195001 (2016).
  31. S. W. Hawking, M. J. Perry, and A. Strominger, Superrotation charge and supertranslation hair on black holes, J. High Energy Phys. 05 (2017) 161.
  32. S. Haco, S. W. Hawking, M. J. Perry, and A. Strominger, Black hole entropy and soft hair, J. High Energy Phys. 12 (2018) 098.
  33. S. Choi, S. Sandeep Pradhan, and R. Akhoury, Supertranslation hair of Schwarzschild black hole: A Wilson line perspective, J. High Energy Phys. 01 (2020) 013.
  34. P. Mao, Gravitational vacua in the Newman-Penrose formalism, Phys. Rev. D 110, 064009 (2024).
  35. A. Strominger, On BMS invariance of gravitational scattering, J. High Energy Phys. 07 (2014) 152.
  36. T. He, V. Lysov, P. Mitra, and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, J. High Energy Phys. 05 (2015) 151.
  37. A. Strominger and A. Zhiboedov, Gravitational memory, BMS supertranslations and soft theorems, J. High Energy Phys. 01 (2016) 086.
  38. A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory (Princeton University Press, Princeton, NJ, 2018).
  39. S. D. Mathur, A. Saxena, and Y. K. Srivastava, Constructing “hair” for the three charge hole, Nucl. Phys. B680, 415 (2004).
  40. I. Bena and N. P. Warner, Bubbling supertubes and foaming black holes, Phys. Rev. D 74, 066001 (2006).
  41. P. Berglund, E. G. Gimon, and T. S. Levi, Supergravity microstates for BPS black holes and black rings, J. High Energy Phys. 06 (2006) 007.
  42. I. Bena, C.-W. Wang, and N. P. Warner, Mergers and typical black hole microstates, J. High Energy Phys. 11 (2006) 042.
  43. I. Bena, C.-W. Wang, and N. P. Warner, Plumbing the abyss: Black ring microstates, J. High Energy Phys. 07 (2008) 019.
  44. J. de Boer, S. El-Showk, I. Messamah, and D. Van den Bleeken, A bound on the entropy of supergravity?, J. High Energy Phys. 02 (2010) 062.
  45. O. Lunin, S. D. Mathur, and D. Turton, Adding momentum to supersymmetric geometries, Nucl. Phys. B868, 383 (2013).
  46. S. Giusto and R. Russo, Superdescendants of the D1D5 CFT and their dual 3-charge geometries, J. High Energy Phys. 03 (2014) 007.
  47. I. Bena, S. Giusto, R. Russo, M. Shigemori, and N. P. Warner, Habemus superstratum! A constructive proof of the existence of superstrata, J. High Energy Phys. 05 (2015) 110.
  48. I. Bena, E. Martinec, D. Turton, and N. P. Warner, Momentum fractionation on superstrata, J. High Energy Phys. 05 (2016) 064.
  49. I. Bena, S. Giusto, E. J. Martinec, R. Russo, M. Shigemori, D. Turton, and N. P. Warner, Smooth horizonless geometries deep inside the black-hole regime, Phys. Rev. Lett. 117, 201601 (2016).
  50. M. Mirbabayi and M. Porrati, Dressed hard states and black hole soft hair, Phys. Rev. Lett. 117, 211301 (2016).
  51. B. Gabai and A. Sever, Large gauge symmetries and asymptotic states in QED, J. High Energy Phys. 12 (2016) 095.
  52. C. Gomez and R. Letschka, Memory and the infrared, J. High Energy Phys. 10 (2017) 010.
  53. Y. Hamada, M.-S. Seo, and G. Shiu, Large gauge transformations and little group for soft photons, Phys. Rev. D 96, 105013 (2017).
  54. R. Bousso and M. Porrati, Soft hair as a soft wig, Classical Quantum Gravity 34, 204001 (2017).
  55. R. Bousso and M. Porrati, Observable supertranslations, Phys. Rev. D 96, 086016 (2017).
  56. G. Compère, J. Long, and M. Riegler, Invariance of Unruh and Hawking radiation under matter-induced supertranslations, J. High Energy Phys. 05 (2019) 053.
  57. F.-L. Lin and S. Takeuchi, Hawking flux from a black hole with nonlinear supertranslation hair, Phys. Rev. D 102, 044004 (2020).
  58. T. M. Adamo, C. N. Kozameh, and E. T. Newman, Null geodesic congruences, asymptotically flat space-times and their physical interpretation, Living Rev. Relativity 12, 6 (2009).
  59. P. Mao and W. Zhao, Twisting asymptotic symmetries and algebraically special vacuum solutions, J. High Energy Phys. 03 (2024) 166.
  60. C. W. Misner, Taub-Nut space as a counterexample to almost anything, in Relativity Theory and Astrophysics. Vol. 1: Relativity and Cosmology, edited by J. Ehlers (American Mathematical Society, Providence, Rhode Island, 1967), Vol. 8, p. 160.
  61. G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105, 111103 (2010).
  62. G. Barnich and C. Troessaert, Supertranslations call for superrotations, Proc. Sci. CNCFG2010 (2010) 010.
  63. G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, J. High Energy Phys. 05 (2010) 062.
  64. U. Kol and M. Porrati, Properties of dual supertranslation charges in asymptotically flat spacetimes, Phys. Rev. D 100, 046019 (2019).
  65. H. Godazgar, M. Godazgar, and C. N. Pope, Dual gravitational charges and soft theorems, J. High Energy Phys. 10 (2019) 123.
  66. E. Newman and R. Penrose, An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. (N.Y.) 3, 566 (1962).
  67. H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, and E. Herlt, Twisting vacuum solutions, in Exact Solutions of Einstein’s Field Equations (Cambridge University Press, Cambridge, England, 2003).
  68. E. T. Newman and T. W. J. Unti, Behavior of asymptotically flat empty spaces, J. Math. Phys. (N.Y.) 3, 891 (1962).
  69. D. W. Trim and J. Wainwright, Nonradiative algebraically special space-times, J. Math. Phys. (N.Y.) 15, 535 (1974).
  70. R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237 (1963).
  71. C. Cherubini, D. Bini, S. Capozziello, and R. Ruffini, Second order scalar invariants of the Riemann tensor: Applications to black hole space-times, Int. J. Mod. Phys. D 11, 827 (2002).
  72. R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128, 2851 (1962).
  73. M. Henneaux and C. Teitelboim, Asymptotically anti-de Sitter spaces, Commun. Math. Phys. 98, 391 (1985).
  74. J. D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity, Commun. Math. Phys. 104, 207 (1986).
  75. G. Barnich, P. Mao, and R. Ruzziconi, BMS current algebra in the context of the Newman–Penrose formalism, Classical Quantum Gravity 37, 095010 (2020).
  76. H. Godazgar, M. Godazgar, and M. J. Perry, Hamiltonian derivation of dual gravitational charges, J. High Energy Phys. 09 (2020) 084.
  77. M. Banados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484, 147 (1999).
  78. H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez, D. Tempo, and R. Troncoso, Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93, 101503 (2016).
  79. M. M. Sheikh-Jabbari and H. Yavartanoo, On 3D bulk geometry of Virasoro coadjoint orbits: Orbit invariant charges and Virasoro hair on locally AdS3 geometries, Eur. Phys. J. C 76, 493 (2016).
  80. M. M. Sheikh-Jabbari, Residual diffeomorphisms and symplectic soft hairs: The need to refine strict statement of equivalence principle, Int. J. Mod. Phys. D 25, 1644019 (2016).
  81. H. Afshar, D. Grumiller, and M. M. Sheikh-Jabbari, Near horizon soft hair as microstates of three dimensional black holes, Phys. Rev. D 96, 084032 (2017).
  82. H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo, and R. Troncoso, Soft hairy horizons in three spacetime dimensions, Phys. Rev. D 95, 106005 (2017).
  83. A. Strominger, Asymptotic symmetries of Yang-Mills theory, J. High Energy Phys. 07 (2014) 151.
  84. T. He, P. Mitra, A. P. Porfyriadis, and A. Strominger, New symmetries of massless QED, J. High Energy Phys. 10 (2014) 112.
  85. S. Chandrasekhar, The Newman-Penrose formalism, in The Mathematical Theory of Black Holes (Oxford University Press, Oxford, United Kingdom, 1983), Chap. 1, pp. 40–55.
  86. J. N. Goldberg and R. K. Sachs, A theorem on Petrov types, Acta Phys. Pol. B 22, 13 (1962).
  87. E. T. Newman and R. Penrose, Note on the Bondi-Metzner-Sachs group, J. Math. Phys. (N.Y.) 7, 863 (1966).

Outline

Information

Phys. Rev. D 113, 024037– Published 20 January, 2026
  • Received 19 July 2025
  • Accepted 5 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation